Good upholstered furniture has zippers on the cushions so you can take the covers off and toss them in the wash. If the cushions themselves are the problem then you can replace them. Depending on what they are made of you may be able to soak them with some laundry detergent and a little bit of bleach, then squeeze them out and let them dry over several days.

For the parts that can't be washed like that, probably the best way to clean them is with a wet vacuum process. You can use a shop-vac for this or one of the rug cleaners with attachments. I have a Hoover steam vac that I use with their allergen reducing cleaner, and I fill the water tank with boiling water.

If that doesn't work then all you can do is have it re-upholstered or trash it.

My mom had an expensive piece with down cushions that she replaced, and gave the cushions to my wife to sew into jackets or cushions. When I brought them into the house she quickly started having allergic reactions so out they went to the compost pile.

I reupholstered a mid 60's car one time and both front seats had that type of foam, but the seat backs and rear sets did not. The upholster told me that they used an organic foam back them (latex?) on critical areas only since it was more expensive then the "cheap" foams.

As I recall I had to wash the covers twice after I soaked them in a weak bleach solution. Although I got the smell out it came back about a year later. I should have replaced the fabric.